You can't fantasize in this way about the historical evolution of notation, Werg22!

To give you just a hint:
The notation f(x) was not at all developed at the time Leibniz chose his notation!
In fact, it was Euler, about 100 years after Leibniz who developed a proto-notation that eventually developed into f(x) in the 19th century.

Furthermore, proofs were generally something very different in the 17th century from proofs of today.

Apart from some solid arguments, most accepted proofs at that time is regarded as mere hand-waving in our time.

There are too many ways that a notation COULD have developed, making it IMPOSSIBLE to deduce how it actually came about.
The only way to find this out is to read the actual works by Leibniz, or better, works by competent commentators on Leibniz.